Quadratic Formula Calculator
Enter the values the quadratic formula form asks for and the result is computed here, in the page, with no sign-up and no upload. Change one field at a time to see how sensitive the quadratic formula figure is to that assumption.
Enter Quadratic Equation Coefficients
Quadratic Formula
| Component | Formula | Description |
|---|---|---|
| Standard Form | ax² + bx + c = 0 | General quadratic equation (a ≠ 0) |
| Discriminant | Δ = b² - 4ac | Determines the nature of roots |
| Quadratic Formula | x = (-b ± √Δ) / 2a | Solution for finding roots |
| Vertex X | x = -b / 2a | X-coordinate of parabola vertex |
| Vertex Y | y = f(-b/2a) | Y-coordinate (minimum or maximum) |
| Axis of Symmetry | x = -b / 2a | Vertical line through vertex |
Understanding the Discriminant
Δ > 0 (Positive Discriminant)
The equation has two distinct real roots. The parabola crosses the x-axis at two points.
Δ = 0 (Zero Discriminant)
The equation has one repeated real root (double root). The parabola touches the x-axis at exactly one point (the vertex).
Δ < 0 (Negative Discriminant)
The equation has two complex conjugate roots. The parabola does not intersect the x-axis. Roots are in the form: a ± bi
Real-World Applications
- Physics: Projectile motion, trajectory calculations, acceleration problems
- Engineering: Structural design, optimization problems, signal processing
- Economics: Profit maximization, cost analysis, break-even points
- Geometry: Area and volume calculations, curve fitting
- Computer Graphics: Collision detection, path planning, animation curves
Use the scientific calculator with steps
Solve ax² + bx + c = 0
Enter a, b, and c to calculate the discriminant and roots. The coefficient a cannot be zero because that changes the problem into a linear equation. The calculator reports when the discriminant is negative instead of hiding the complex-root case.
Formula
The quadratic formula is x = (-b plus or minus the square root of b² minus 4ac) divided by 2a. The discriminant b² minus 4ac tells you whether there are two real roots, one repeated real root, or complex roots.
Worked example
For x² minus 5x plus 6 = 0, the discriminant is 1 and the roots are 2 and 3. Substitute each root into the original equation to check the result.
Frequently Asked Questions about Quadratic Formula Calculator
How do I sanity-check the quadratic formula result?
Rerun the calculator with a known example and compare it against the figure you expect; then swap back to your own numbers.
Which quadratic formula inputs change the outcome most?
Adjust one field at a time. The figure that shifts the result hardest is the assumption worth double-checking before you rely on it.
What if a equals zero?
Use a linear equation method because the quadratic formula no longer applies.
What does a negative discriminant mean?
It means there are no real roots and the solutions are complex.
Are the steps symbolic proof?
The page shows numeric formula results and should be checked against the original equation.